Constructibility and Mathematical Existence

Chihara, Charles S.

doi:10.1093/0198239750.001.0001

Cited by 39 publications

(41 citation statements)

References 45 publications

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“…I argue that even given the sort of explanation most favorable to such inference, typical explanations of physical facts involving mathematics do not commit us to mathematical objects. As such the position will be consistent with antirealist views that understand mathematical truths as modal claims of possibility, in line with the suggestion of Putnam (2012), and the accounts proposed by Chihara (1990) and Hellman (1989).…”

Section: Baker's Indispensability Argument and Strategies Of Responsesupporting

confidence: 83%

Mathematical explanation and indispensability

Vineberg

2018

THEORIA

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This paper discusses Baker's Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.Keywords: indispensability, explanation, realism, structuralism.RESUMEN: Este artículo analiza la versión 'mejorada' del argumento de la indispensabilidad (EIA) de Baker y varias de las respuestas que ha recibido. Este argumento pretende establecer el realismo matemático en virtud del papel indispensable que las matemáticas juegan en las explicaciones científicas de los hechos físicos. Argumento que hay un análogo de explicación causal para las matemáticas que, de los varios sentidos básicos de explicación, parece el más adecuado para el EIA. Analizo un caso plausible en el que las matemáticas juegan un papel explicativo en este sentido, pero concluyo que este uso no es suficiente para establecer el realismo acerca de los objetos matemáticos.

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Section: Baker's Indispensability Argument and Strategies Of Responsesupporting

confidence: 83%

Mathematical explanation and indispensability

Vineberg

2018

THEORIA

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“…But note that some systems with non-standard logical terms, e.g., first-order logic with the quantifier "there exist uncountably many," are also complete. (See Keisler [19] Thus, take a simple interpreted language, Z, and consider the following three inference schemata of Z: (6) a is yellow (all yellow); therefore, a is not red, where a is to be replaced by an individual term of Z. To distinguish necessary from non-necessary inferences of Z, the apparatus of models for Z has to be constructed in such a way that all instances of (6) and (8) preserve truth in all models while some instances of (7) do not.…”

Section: Does (5) Hold In Standard Semantics?mentioning

confidence: 99%

Did Tarski commit “Tarski's fallacy”?

Sher

1996

J. symb. log.

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In his 1936 paper, On the Concept of Logical Consequence, Tarski introduced the celebrated definition of logical consequence: “The sentenceσ follows logically from the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentence σ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentence σ is false. Moreover, … we are concerned here with the concept of logical, i.e., formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since.

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“…Over the last century, a variety of schools and point of views has seen rise about the meaning of existence, definability and conceivability of mathematical objects (Field 1980;Chihara 1990;Burgess and Rosen 1997;Lakoff and Nunez 2000). A much-debated object is that of a "number" (Benacerraf 1965;Frege 1983).…”

Section: Defining and Building New Objects: A Common Foundational Conmentioning

confidence: 99%

On the imaginative constructivist nature of design: a theoretical approach

Kazakçı

2013

Res Eng Design

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Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in current design research. By contrast, in other fields such as Mathematics or Psychology, the notion of constructivism is seen as a foundational issue. The present paper defends the point of view that forms of constructivism in design need to be identified and integrated as a foundational element in design research as well. In fact, a look at the literature reveals at least two types of constructive processes that are well embedded in design research. First, an interactive constructivism, where a designer engages a conversation with media, that allows changing the course of the activity as a result of this interaction. Second, a social constructivism, where designers need to handle communication and negotiation aspects, that allows integrating individuals' expertise into the global design process. A key feature lacking to these well-established paradigms is the explicit consideration of creativity as a central issue of design.To explore how creative and constructivist aspects of design can be taken into account conjointly, the present paper pursues a theoretical approach. We consider the roots of constructivism in mathematics, namely, the Intuitionist Mathematics, in order to shed light on the original insights that led to the development of a notion of constructivism. Intuitionists describe mathematics as the process of mental mathematical constructions realized by a creative subject over time. One of the most original features of Intuitionist Constructivism is the introduction of incomplete objects into the heart of mathematics by means of lawless sequences and free choices. This allows the possibility to formulate undecided propositions and the consideration of creative acts within a formal constructive process. We provide an in-depth analysis of Intuitionism from a design standpoint showing that the original notion is more than a pure constructivism where new objects are a mere bottom-up combination of already known objects. Rather, intuitionism describes an imaginative constructivist process that allows combining bottom-up and top-down processes and the expansion of both propositions and objects with free choices of a creative subject. We suggest that this new form of constructivism we identify is also relevant in interpreting conventional design processes and discuss its status with respect to other forms of constructivism in design. Keywords Centre de Gestion Scientifique, Design Chair, MINES PARISTECHAbstract: Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in current design research. By contrast, in other fields such as Mathematics or Psychology, the notion of cons...

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Constructibility and Mathematical Existence

Cited by 39 publications

References 45 publications

Mathematical explanation and indispensability

Mathematical explanation and indispensability

Did Tarski commit “Tarski's fallacy”?

On the imaginative constructivist nature of design: a theoretical approach

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